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Chapter 3: Problem 15

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{2}\right)^{x}$$

Short Answer

Expert verified
The graph of the function \(h(x) = (\frac{1}{2})^x\) starts from the point (-2,4) and gradually curls downwards as x increases. The graph decreases as x increases, in line with the property of exponential functions with a base less than 1.

Step by step solution

01

- Creating a table of coordinates

Choose a couple of x-values and substitute them into the function \(h(x) = (\frac{1}{2})^x\) and calculate the corresponding y-values. It is a good choice to select a spread of positive, negative and zero values for x. For instance, consider x-values: -2, -1, 0, 1, and 2. The corresponding y-values are: \(h(-2) = (\frac{1}{2})^{-2} = 4\), \(h(-1) = (\frac{1}{2})^{-1} = 2\), \(h(0) = (\frac{1}{2})^{0} = 1\), \(h(1) = (\frac{1}{2})^{1} = \frac{1}{2}\), \(h(2) = (\frac{1}{2})^{2} = \frac{1}{4}\)
02

- Plot the coordinates on the graph

Take your x and y coordinates, and plot them on the Cartesian plane. The points to plot from our table are (-2,4), (-1,2), (0,1), (1,1/2) and (2,1/4). Connect these points, and note that the graph should be a curve that decreases as x increases.
03

- Confirming the graph using a graphing utility

Enter the function \(h(x) = (\frac{1}{2})^x\) into the graphing utility and compare the hand-drawn graph to the output of the graphing utility. This cross-verification will help to confirm the correctness of the manually drawn graph.

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